Range of an equation

Question

I am trying to find the range of: $\Large\frac{4 - {\frac {4 + a^2}{a}}}{\frac {1}{2} - \frac {1}{a}}$

So, we have: $a \ne 0$

$\frac {1}{2} - \frac {1}{a} \ne 0 $, $-\frac {1}{a} \ne -\frac {1}{2}$, $\frac {1}{a} \ne \frac {1}{2}$

I am not sure how to solve the last inequality. Thank you in advance!

I would first simplify the expression into a non-rational form. Then find the domain. And then "transform" the domain to the range. — RayDansh, Jun 29 '19 at 19:53

score 3 · Accepted Answer · answered Jun 29 '19 at 20:05

3

Simplify it and you get $-2(a-2)=4-2a$. With $a \neq 0 \implies 4-2a \neq 4$, and for $a \neq 2 \implies 4-2a \neq 0$. Thus the range is $(-\infty, 0)\cup (0,4)\cup (4,\infty)$.

answered Jun 29 '19 at 20:05

DeepSea

77,651

score 1 · Answer 2 · answered Jun 29 '19 at 19:31

1

It is equivalent to $$\frac{-\frac{(a-2)^2}{a}}{\frac{a-2}{2a}}$$

answered Jun 29 '19 at 19:31

Dr. Sonnhard Graubner

95,283

Range of an equation

2 Answers2